# Which group actions are "set-transitive"?

Given a faithful group action $$G$$ on a finite set $$X$$ of cardinality $$n$$ and an integer $$1\le k\le n$$, say that $$G$$ is $$k$$-set-transitive if we can map any unordered $$k$$-tuple in $$X$$ to any other via an element $$g\in G$$. (Contrast with being $$k$$-transitive, in which case this holds for ordered tuples.) Say that $$G$$ is set-transitive if it is $$k$$-set-transitive for all $$k$$.

Obviously, $$S_n$$ and $$A_n$$ are set-transitive with their natural action on an $$n$$-element set (except $$n=2$$ in the alternating case), and in general $$k$$-transitivity implies both $$k$$-set-transitivity and $$(n-k)$$-set-transitivity. The Mathieu group $$M_{12}$$ is $$k$$-set-transitive for all $$k\neq 6$$.

Some questions:

• Are there any set-transitive actions besides the two categories mentioned above?

• Besides $$A_4$$ and $$A_5$$, are there any groups which are $$k$$-set-transitive for all $$k\le 4$$ which are not also $$4$$-transitive?

• More generally, are there any group actions which are $$k$$-set-transitive for some $$k\le n/2$$ but not $$k$$-transitive? The condition seems much weaker, but I'm struggling to generate examples.

Any pointers to discussion of this property in the literature would also be welcome.

The standard term for a group that is transitive on $$k$$-sets is $$k$$-homogeneous.

A basic result is due to Livingstone and Wagner (1965) and Kantor (1972):

Suppose that $$G \le {\rm Sym}(X)$$ with $$|X| \le n$$, and assume that $$G$$ is $$k$$-homogeneous with $$k \le n/2$$ (that is no restriction since $$G$$ is $$k$$-homogeneous if and only if it is $$(n-k)$$-homogeneous). Then $$G$$ is $$(k-1)$$-transitive and, with the following exceptions, $$G$$ is $$k$$-transitive:

(i) $$k=2$$, $${\rm ASL}_1(q) \le G \le {\rm A \Sigma L}_1(q)$$ with $$q \equiv 3 \bmod 4$$;

(ii) $$k=3$$, $${\rm PSL}_2(q) \le G \le {\rm P \Sigma L}_2(q)$$ with $$q \equiv 3 \bmod 4$$;

(iii) $$k=3$$, $$G={\rm AGL}_1(8)$$, $${\rm A \Gamma L}_1(8)$$, or $${\rm A \Gamma L}_1(32)$$;

(iv) $$k=4$$, $$G={\rm PGL}_2(8)$$, $${\rm P \Gamma L}_2(8)$$, or $${\rm P \Gamma L}_2(32)$$.

This is stated (but not proved) as Theorem 9.4B in Dixon & Mortimer's book "Permutation Groups".